Jude Law lands on deck, flung from the ocean like a helpless fish. He's been on board only a few minutes before Ruth Wilson gives him a knockout punch: it looks so convincing that for a moment it seemed that first-nighters were going to be present at one of those theatrical accidents in which a pretend injury turns into a real-life disaster. Soon an iron bedstead on which Wilson is sprawled is lifted and slammed down as if it were a matchbox, and Law is heaving up Wilson's old sea-dog dad to carry him horizontal and chest-high like an awkward parcel. When the floor of a bar tips to become the deck of a coal barge, it is raked as steeply as a roof.

Rob Ashford's production of Anna Christie is pell-mell, tumultuous, full of blow and go. It has in Law and Wilson two actors giving exposing, outsize performances which secure them as stage actors. Wilson is the young woman of the title: come to seek the bargee father whom she hasn't seen since she was a tot, she conceals for a time that she has been a prostitute, then lets her father and new lover have the truth with caustic anger and complicated results.

It's a role pitted with sentimental pitfalls but Wilson triumphs in it: strutting and slumping like a sot but talking in languorous cadences which float questioningly over her unflinching statements. When she tugs at the clouds of fog that swirl across the stage it's as if she were trying to clear her mind. In the part of the Irish stoker who falls in love with her, Law is physically and linguistically bulked up: his accent is as thick as Kerrygold; his arms dangle like a boxer's; his desire and rage send shudders around and before him.

Eugene O'Neill's play, first staged in 1921, needs this full-throttle approach, with acting that punches as well as inflects. It courts melodrama and has stiff passages. As the gnarled, neglectful father, David Hayman bears the brunt of O'Neill's lapses into absurdity. His fighting-cock stance and wary gaze exactly capture the defensiveness of the guilty escaper, but his inflated lines – often railing against "dat old deffil sea" – are delivered in an accent, supposedly Swedish, that is so strangulated and peculiar it could be Elvish.

A daring velocity overwhelms, though not quite banishes, these difficulties. There is a shipwreck, there is lashing rain, there is encompassing fog. Adam Cork, whose music and soundscapes have become such a vital feature of the stage over the past decade, marinates the action in a keening sea shanty and the distant ring of bells: the evening has the salty flavour of tears as well as the ocean. In the 1930 film of Anna Christie, Greta Garbo made her speaking debut with the gorgeous throwaway: "give me a vhisky, ginger ale on the side… and don't be stingy, baby". Ashford's production has the shot and the fizz, and it ain't stingy.

"Never was space ere so speedily spanned," cackles Lucifer in The Globe Mysteries. He's right. Tony Harrison has scaled down his 1985 epic to a trim three hours, which patters in Deborah Bruce's production from creation (Adam and Eve step out of packing cases) through the fall (forbidden fruit drops from a tree of balloons) and the crucifixion, with William Ash's delicate, clear-spoken Christ hoisted high on a terrifyingly thick slab of wood, to the day of judgment, where – the Globe coming into its own – the groundlings are divided between damned and saved, with a shifty cleric trying to slip from one side to the other.

Fifteen actors take on some 60 parts in a production which is quick and comic. God is not so much wrathful as cross; Lucifer is at least as vain as he is proud, pulling down his vest to show his name tattooed on his puffed-out chest; Mrs Noah is so keen on finishing her knitting that she has to be lifted into the ark; one of the executioners has to be helped out when he hits his thumb while hammering in a nail.

It would be more far-reaching if it put more trust in the capacity of Harrison's vernacular to express tragedy, if it contained more dark and quiet moments – Mary's address to the cross is finely rendered – more music (there is one outstanding female voice who could be a guiding thread) and fewer mobile-phone gags. Yet the strength of Harrison's verse shines through: the sprightly rhyme and the encrustations of alliteration make it at once catchy and detaining. You have to attend to the force of his meaning.

Out of the shadows of Regent's Park comes a chorus of high-kickers in turquoise spangled bodices: they are like moonbeams, scattering the darkness as they dance in silver light then twinkle off, leaving Sean Palmer to tap his dextrous way across the stage in a violet haze.

Timothy Sheader's glorious production of Crazy for You also began in cloud (instead of that eerie pause before the music starts, there was a concerted gasp, as the rain pelted) and went on to shine. Literally. One of the splendours of the Open Air theatre since Sheader took over has been lighting (here by Tim Mitchell) which sweeps the stage with moods and transports the audience from their damp seats.

Ken Ludwig's 1992 makeover of George and Ira Gershwin's 1930 musical Girl Crazy, which blithely gathers in Gershwin songs from all over the place, contains not only "I Got Rhythm" but "Embraceable You". Its most captivating moments are choreographic. Stephen Mear's skilful peak comes with "Slap That Bass", in which scattered talents slowly build into a one-beat ensemble, but there are continual mini-peaks throughout the evening. There's the song of two drunks – one dressed up as the other – who slither, jump, glug and slump together in such perfect unison that when they come eyeball to eyeball each thinks he might be seeing double: "I am beside myself," slurs one. There's the trio of cowboys who come on bearing a huge plastic cactus and wheeze out lines about laziness with a sublime ease. There's the immense bad girl number and the swooning good girl song. Hard not to be crazy for it.